Polynomial imaginary decompositions for finite extensions of fields of characteristic zero
Andrzej Nowicki, Stanisław Spodzieja · Bulletin of the Polish Academy of Sciences Mathematics · 2003
Let k be a field of characteristic zero, L = k[ξ] a finite field extension of degree m > 1, and f(z) a polynomial in one variable z over L. Then there exist unique polynomials u0, . . . , um−1 belonging to k[x0, . . . , xm−1] such that f(x0 + ξx1 + · · · + ξxm−1) = u0 + ξu1 + · · ·+ ξum−1. We prove that if f(z) 6∈ L, then the polynomials u0, . . . , um1 are algebraically independent over k and they have no common divisors in k[x0, . . . , xm−1] of positive degrees. Some other properties of polynomials u0, . . . , um−1 are also given.